| Antipodal Point |
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In Mathematics , the antipodal point of a point on the surface of a sphere is the point which is Diametrically opposite it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter. An antipodal point is sometimes called an antipode, a Back-formation from the Greek Loan Word ''antipodes'', which originally meant "opposite the feet." THEORY In , they are points with related Vector s v and −v. On a Circle , such points are also called '''diametrically opposite'''. In other words, each line through the centre intersects the sphere in two points, one for each Ray out from the centre, and these two points are antipodal. The Borsuk-Ulam Theorem is a result from Algebraic Topology dealing with such pairs of points. It says that any Continuous Function from ''S''''n'' to R''n'' maps a pair of antipodal points in ''S''''n'' to the same point in R''n''. Here, ''S''''n'' denotes the sphere in ''n''-dimensional space (so the "ordinary" sphere is ''S''3). The antipodal map ''A'' : ''S''''n'' → ''S''''n'', defined by ''A''(''x'') = −''x'', sends every point on the sphere to its antipodal point. It is Homotopic to the Identity Map if ''n'' is odd, and its Degree is (−1)''n''+1. If one wants to consider antipodal points as identified, one passes to Projective Space (see also Projective Hilbert Space , for this idea as applied in Quantum Mechanics ). REFERENCES EXTERNAL LINKS |
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